GCSE Maths Probability revision

Probabilities are numbers from 0 to 1: 0 means impossible, 1 means certain, 0.5 means evens. For equally likely outcomes, P(event) = favourable outcomes ÷ total outcomes: a bag with 3 red, 5 blue and 2 green counters gives P(red) = 3⁄10 = 0.3. Fractions, decimals and percentages are all legal clothing — 0.25, ¼ and 25% are the same bet.

The complement shortcut halves the work: P(not A) = 1 − P(A). If P(win) = 0.15 then P(not win) = 0.85, and “not win” covers every other outcome at once. For a set of mutually exclusive events — outcomes that cannot happen together — probabilities add: P(rain) + P(snow) with no simultaneous rain-and-snow covers the wet family completely. When the listed events are also exhaustive, the total is exactly 1; anything missing belongs to “neither”.

Theory prices an event; experiments test the price. Flip a fair coin 500 times and heads might land 228 times: the relative frequency is 228⁄500 = 0.456, close to (but not equal to) the theoretical 0.5. Large samples settle towards theory — a 10-flip experiment proving 80% heads says nothing. Comparing the two is how GCSE questions ask you to judge whether a die or spinner is fair; a big gap over many trials suggests bias.

Predictions run the multiplication backwards and forwards: expected frequency = probability × number of trials. A spinner with P(win) = 0.22 spun 400 times should win about 0.22 × 400 = 88 times. Expected values can be decimals or fractions — they are averages over many repeats, not promises about any single run. A game is fair when each player’s expected score is equal.

A Venn diagram sorts a sample set ξ into overlapping loops. Fill the intersection (both) first, then subtract outwards to the only-regions, and put the leftover outside both loops. Thirty students with 18 studying maths, 15 physics and 7 both give: only maths 11, only physics 8, both 7, neither 4 — the counts sum to 30, which is the check. The union rule n(A ∪ B) = n(A) + n(B) − n(A ∩ B) does the same job in symbols: 18 + 15 − 7 = 26.

Formal set notation (Higher) compresses the language: A ∪ B is A or B (union), A ∩ B is A and B (intersection), A′ is not A, ∈ means is a member of, ∉ is not, ∅ is the empty set, and n(A) counts A’s members. A frequency tree splits a population stage by stage — 80 pupils, 3⁄5 boys, a quarter of boys wearing glasses, three eighths of girls — and every branch question is one division. Two-way tables carry the same data in rows and columns; conditional questions (Higher) divide a cell by its row total.

A tree diagram lays out every path of a combined experiment. Two rules run it: multiply along a path (both events happening), and add between distinct paths (this way or that way). For two coins, P(HH) = ½ × ½ = ¼. For a bag of 4 red and 6 blue with replacement, the second draw copies the first; without replacement the denominators shrink: P(R then B) = 4⁄10 × 6⁄9 = 4⁄15. Forgetting to shrink the second fraction is the classic tree-diagram error.

Conditional probability (Higher) asks: given the first branch happened, what now? From the same bag, P(second is blue | first was red) = 6⁄9 = ⅔ — the condition redraws the denominator. Two-way tables and Venn diagrams hide the same division: P(driver | woman) is the women-who-drive cell over the whole women row. In set language, P(A | B) counts A ∩ B over n(B).